Computation of Steady Incompressible Flows in Unbounded Domains

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Abstract

In this study we revisit the problem of computing steady Navier-
Stokes flows in two-dimensional unbounded domains. Precise quanti-
tative characterization of such flows in the high-Reynolds number limit
remains an open problem of theoretical fluid dynamics. Following a
review of key mathematical properties of such solutions related to the
slow decay of the velocity field at large distances from the obstacle,
we develop and carefully validate a spectrally-accurate computational
approach which ensures the correct behavior of the solution at infin-
ity. In the proposed method the numerical solution is defined on the
entire unbounded domain without the need to truncate this domain
to a finite box with some artificial boundary conditions prescribed at its boundaries. Since our approach relies on the streamfunction-
vorticity formulation, the main complication is the presence of a dis-
continuity in the streamfunction field at infinity which is related to
the slow decay of this field. We demonstrate how this difficulty can be
overcome by reformulating the problem using a suitable background
”skeleton” field expressed in terms of the corresponding Oseen flow
combined with spectral filtering. The method is thoroughly validated
for Reynolds numbers spanning two orders of magnitude with the re-
sults comparing favourably against known theoretical predictions and
the data available in the literature.

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